Cheap thermocouples. Homemade

My understanding is that the thermocouple voltage is strictly a function of the temperature. Apparently this is called the Seebeck Voltage. Like any voltage source there is an internal resistance associated with it due to "real world" effects. The short-circuit current will be determined by that internal resistance by the usual Ohm's Law calculation I_ss = V(temperature) / R_internal.

Things you need to decide- whether your junction will be grounded or not. If they are grounded you will need differential measurement with good common-mode rejection. If they are ungrounded you don't need that (assuming low leakage). Advantages are that grounded-junction is usually much faster response and can be cheaper.

Also you need to provide cold-junction compensation. That means that you have to measure the temperature (with something other than a thermocouple!) of where the wires from the thermocouple are connected to copper (usually a screw terminal block). A bit of math is required to do the compensation, depending on the accuracy you need. The error in your cold-junction sensor's reading is about 1:1 with error in your thermocouple reading- meaning if you want accuracy of 1°C you might need to have the terminals isothermal and measure to within a total error of +/-0.5°C. This is non-trivial if the environment isn't rather benign.

Resolution as you calculated is only part of it- you need to make sure that the zero drift of your amplifier/ADC is low enough. The output of a type K thermocouple is not insanely low, but you need to be fairly careful. For example, a cheap op-amp could easily drift that much with a few degrees C change and could have 50 or 100 degrees initial error due to the offset.

Remember, you must use thermocouple or thermocouple extension wire and appropriate connectors all the way to your measuring instrument, where the cold-junction compensation takes place. A thermocouple is not just like a battery supplying a fixed voltage for temperature. The temperature of the connection points matters about as much as the temperature of the measuring junction!